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Spectral invariance, ellipticity, and the Fredholm property for pseudodifferential operators on weighted Sobolev spaces

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Abstract

The pseudodifferential operators with symbols in the Grushin classes \~S supρ,δinf0 , 0 ≤ δ < ρ ≤ 1, of slowly varying symbols are shown to form spectrally invariant unital Frécher-*-algebras (Ψ*-algebras) in L(L 2(Rn)) and in L(H γ st) for weighted Sobolev spaces H supstinfγ defined via a weight d function γ. In all cases, the Fredholm property of an operator can be characterized by uniform ellipticity of the symbol. This gives a converse to theorems of Grushin and Kumano-Ta-Taniguchi. Both, the spectrum and the Fredholm spectrum of an operator turn out to be independent of the choices of s, t and γ.

The characterization of the Fredholm property by uniform ellipticity leads to an index theorem for the Fredholm operators in these classes, extending results of Fedosov and Hörmander.

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  1. Fachbereich Mathematik, Johannes Gutenberg-Universität, D-6500 Mainz, Germany

    Elmar Schrohe

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  1. Elmar Schrohe
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Communicated by B. W. Schulze

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Schrohe, E. Spectral invariance, ellipticity, and the Fredholm property for pseudodifferential operators on weighted Sobolev spaces. Ann Glob Anal Geom 10, 237–254 (1992). https://doi.org/10.1007/BF00136867

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